reserve i,n,m,k,x,y for Nat,
  i1 for Integer;

theorem Th6:
  2 <= k & m is_represented_by 1,k implies DigA_SDSub(SD2SDSub(
  DecSD(m,1,k)),1) = m - SDSub_Add_Carry(m,k)*Radix(k)
proof
  assume that
A1: 2 <= k and
A2: m is_represented_by 1,k;
A3: 1 in Seg 1 by FINSEQ_1:3;
A4: 1 in Seg (1+1) by FINSEQ_1:1;
  then
  DigA_SDSub(SD2SDSub(DecSD(m,1,k)),1) = SD2SDSubDigitS(DecSD(m,1,k), 1, k
  ) by RADIX_3:def 8
    .= SD2SDSubDigit(DecSD(m,1,k), 1, k) by A1,A4,RADIX_3:def 7;
  hence
  DigA_SDSub(SD2SDSub(DecSD(m,1,k)),1) = SDSub_Add_Data( DigA(DecSD(m,1,k
  ),1), k ) + SDSub_Add_Carry( DigA(DecSD(m,1,k),1-'1), k) by A3,RADIX_3:def 6
    .= SDSub_Add_Data( DigA(DecSD(m,1,k),1), k ) + SDSub_Add_Carry( DigA(
  DecSD(m,1,k),0), k) by XREAL_1:232
    .= SDSub_Add_Data( DigA(DecSD(m,1,k),1), k ) + SDSub_Add_Carry( 0, k) by
RADIX_1:def 3
    .= SDSub_Add_Data( m, k ) + SDSub_Add_Carry( 0, k) by A2,RADIX_1:21
    .= SDSub_Add_Data( m, k ) + 0 by RADIX_3:16
    .= m - SDSub_Add_Carry(m,k) * Radix(k) by RADIX_3:def 4;
end;
